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Theorem dvdsval2 13534
Description: One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
Assertion
Ref Expression
dvdsval2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )

Proof of Theorem dvdsval2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 divides 13533 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
213adant2 1002 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. k  e.  ZZ  ( k  x.  M )  =  N ) )
3 zcn 10647 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
433ad2ant3 1006 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  N  e.  CC )
54adantr 462 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  N  e.  CC )
6 zcn 10647 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  CC )
76adantl 463 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  k  e.  CC )
8 zcn 10647 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
983ad2ant1 1004 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  e.  CC )
109adantr 462 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  e.  CC )
11 simpl2 987 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  M  =/=  0
)
125, 7, 10, 11divmul3d 10137 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  N  =  (
k  x.  M ) ) )
13 eqcom 2443 . . . . . . . 8  |-  ( N  =  ( k  x.  M )  <->  ( k  x.  M )  =  N )
1412, 13syl6bb 261 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( N  /  M )  =  k  <->  ( k  x.  M )  =  N ) )
1514biimprd 223 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  k  e.  ZZ )  ->  ( ( k  x.  M )  =  N  ->  ( N  /  M )  =  k ) )
1615impr 616 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  =  k )
17 simprl 750 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
k  e.  ZZ )
1816, 17eqeltrd 2515 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  ( k  x.  M )  =  N ) )  -> 
( N  /  M
)  e.  ZZ )
1918rexlimdvaa 2840 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  -> 
( N  /  M
)  e.  ZZ ) )
20 simpr 458 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( N  /  M )  e.  ZZ )
21 simp2 984 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  M  =/=  0 )
224, 9, 21divcan1d 10104 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  x.  M )  =  N )
2322adantr 462 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  ( ( N  /  M )  x.  M )  =  N )
24 oveq1 6097 . . . . . . 7  |-  ( k  =  ( N  /  M )  ->  (
k  x.  M )  =  ( ( N  /  M )  x.  M ) )
2524eqeq1d 2449 . . . . . 6  |-  ( k  =  ( N  /  M )  ->  (
( k  x.  M
)  =  N  <->  ( ( N  /  M )  x.  M )  =  N ) )
2625rspcev 3070 . . . . 5  |-  ( ( ( N  /  M
)  e.  ZZ  /\  ( ( N  /  M )  x.  M
)  =  N )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2720, 23, 26syl2anc 656 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  /\  ( N  /  M )  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  M
)  =  N )
2827ex 434 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  (
( N  /  M
)  e.  ZZ  ->  E. k  e.  ZZ  (
k  x.  M )  =  N ) )
2919, 28impbid 191 . 2  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( k  x.  M
)  =  N  <->  ( N  /  M )  e.  ZZ ) )
302, 29bitrd 253 1  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714   class class class wbr 4289  (class class class)co 6090   CCcc 9276   0cc0 9278    x. cmul 9283    / cdiv 9989   ZZcz 10642    || cdivides 13531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-z 10643  df-dvds 13532
This theorem is referenced by:  dvdsval3  13535  nndivdvds  13537  fsumdvds  13572  3dvds  13592  bitsmod  13628  sadaddlem  13658  bitsuz  13666  mulgcd  13726  sqgcd  13738  prmind2  13770  mulgcddvds  13786  qredeu  13789  isprm5  13794  divgcdodd  13801  divnumden  13822  hashdvds  13846  oddprm  13878  pythagtriplem11  13888  pythagtriplem13  13890  pythagtriplem19  13896  pcprendvds2  13904  pcpremul  13906  pc2dvds  13941  pcz  13943  pcadd  13947  pcmptdvds  13952  fldivp1  13955  pockthlem  13962  prmreclem1  13973  prmreclem3  13975  4sqlem8  14002  4sqlem9  14003  4sqlem12  14013  4sqlem14  14015  sylow1lem1  16090  sylow3lem4  16122  odadd1  16323  odadd2  16324  pgpfac1lem3  16568  prmirredlem  17876  prmirredlemOLD  17879  znidomb  17953  root1eq1  22152  atantayl2  22292  efchtdvds  22456  dvdsdivcl  22480  muinv  22492  chtub  22510  bposlem6  22587  lgseisenlem1  22647  lgsquad2lem1  22656  lgsquad3  22659  m1lgs  22660  2sqlem3  22664  2sqlem8  22670  qqhval2lem  26346  nn0prpwlem  28442  congrep  29241  jm2.22  29269  jm2.23  29270  hashgcdlem  29490  proot1ex  29494
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